In this context, are bias and variance always related to MSE, or only when the cost function for the statistical learning method is related to MSE?
I think you're totally confused. Bias/variance relates how well your model fits relative to overfitting and underfitting. MSE is simply one of the many possible measures we can use to quantify how well a model performs.
High bias can be measured by very high MSE and high variance can be measured by very low MSE.